Pattern Databases by Joseph C. Culberson and Jonathan Schaeffer 1 - - PowerPoint PPT Presentation

Pattern Databases by Joseph C. Culberson and Jonathan Schaeffer

Presented by Patrick von Reth 1

The 15-Puzzle

 15-Puzzle is a permutation problem.

 Problems where a set of operators converts one permutation into another  Reach goal permutation with as few operators as possible

 Another prominent example is the Rubik’s Cube.

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The 15-Puzzle

1 5 6 3 12 9 4 2 15 11 10 7 13 14 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Manhattan Distance

 Distance between two positions in a grid  Sum of the vertical and horizontal distance  Example: Distance for the empty tile to its goal position.  Manhattan heuristic: Sum of distances for all tiles to their goal position. 1 5 2 3 4 9 6 7 8 10 11 12 13 14 15

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Linear conflict heuristic

 Extension of the Manhattan heuristic.  If two tiles are in linear conflict they have to surround each other. Increase the heuristic by two.  Example: The Manhattan distance for tile 4 and for tile 7 is two. 1 2 3 5 7 4 6 8 9 10 11 12 13 14 15

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Pattern databases

 Human-like approach to solve a complex permutation problems:

 Find a partial solution and thus reduce the complexity of the problem.

 Find an optimal path to a partial solution  Distance to the partial solution is a lower bound for the complete solution.  Precompute the distance to the partial solution, for all permutations, and store it in a database.

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Fringe and Corner pattern

 The Corner pattern  The Fringe pattern 8 9 10 12 13 14 15 3 7 11 12 13 14 15

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Upper bounds

 Used to eliminate the last iteration of the IDA* search.  Upper bound for the 15-Puzzle

 For the fringe pattern the upper bound is 61 moves.  For the remaining 8-Puzzle the upper bound is 31 moves.  This results in an upper limit of 92 moves to completely solve the 15-Puzzle.

 Different patterns result in different upper bounds.

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Solution Databases

 If obtaining the optimal solution path is to expensive.

 Precompute optimal solution for sub-goals together with the moves.

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Symmetry

 Can be used to simplify the search tree.  For example reflections can be used to obtain additional cost bounds.

 The maximum of all lower bounds retrieved is then used.  Two diagonal reflections , (0,15) and (12,3).  The vertical and the horizontal reflection.

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Symmetry - Reflections

 Applying the diagonal reflection (0,15) on a path yields the mirrored path.  Can be achieved by remapping the operators

 d->r, r->d, l->u, u->l

 Used to retrieve the cost for the reflected state.  With some modifications this can also be done for the other reflections.

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Symmetry - Cost retrieved using reflections

 Goal state and its horizontal mirror.

 Distance of the two state is three.  Distance of three is used as penalty on the retrieved cost for the mirror state.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Symmetry – Reflection penalties

 Cost for diagonal reflection (0,15) and its mirrored state are the same.  Penalty for the horizontal and the vertical reflection is 3.  Penalty for the diagonal (12,3) reflection is 6.

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Combination of patterns

 Both pattern databases are used for the heuristic calculation.

 Using maximum over both databases would be expensive (memory)  Split up in 16 databases each.  Lookup defined by empty tile and its mirrors.  Only one half of each database is loaded.

F C F C C F C F F C F C C F C F

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Linear conflict heuristic compared to pure Manhattan distance heuristic

 Presented results are based on the Manhattan distance.

 Linear conflict heuristic:

 Search tree size is reduced noticeable.

 Pattern database with linear conflict heuristic:

 Only minor improvement.

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Results (1)

 Combination of both pattern databases is better than individual improvements.  Many enhancements possible

 Don’t use the diagonal (12,3) because of the penalty of 6.  Don’t lookup every possible reflection.

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Results (2)

 Without those improvements:

 12 times faster than a naïve implementation only using the Manhattan distance.  1.5 times faster than using the linear conflict heuristic

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Conclusion

 Computational workload increased.

 Computation is done before the actual search.  Computation is done once, cost will amortise over time.

 Speedup of 12 and reduction of the search tree size by 1038-fold is impressive.  Pattern databases are better than the linear conflict heuristic.  Adaptable for different problems.

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Questions ?

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